AP Calculus Unit 6 MCQ Review

Front

What is the definition of a derivative?

Back

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. It represents the slope of the tangent line to the graph of the function at that point.

Front

What is the Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on [a, b], then the integral of its derivative over that interval equals the difference in the values of the function at the endpoints: ∫_a^b f'(x) dx = f(b) - f(a).

Front

How do you find the critical points of a function?

Back

Critical points occur where the derivative is zero or undefined. To find them, set the derivative of the function equal to zero and solve for x.

Front

What is the Mean Value Theorem?

Back

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes. This occurs when the second derivative is zero or undefined and changes sign.

Front

What is the difference between a local maximum and a local minimum?

Back

A local maximum is a point where the function value is higher than all nearby points, while a local minimum is a point where the function value is lower than all nearby points.

Front

What is the purpose of the first derivative test?

Back

The first derivative test is used to determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the derivative before and after the critical point.

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function at a critical point. If f''(c) > 0, the function has a local minimum at c; if f''(c) < 0, it has a local maximum; if f''(c) = 0, the test is inconclusive.

Front

What is a limit?

Back

A limit is a value that a function approaches as the input approaches a certain point. It is fundamental in defining derivatives and integrals.

Front

What is the definition of continuity at a point?

Back

A function is continuous at a point x = c if the following three conditions are met: f(c) is defined, the limit of f(x) as x approaches c exists, and the limit equals f(c).